A general algorithm for obtaining simple structure of core arrays in N-way PCA with application to fluorometric data

Simplifying the structure of core arrays from N-way PCA or Tucker3 models is desirable to allow for easy interpretation of the factor estimates. In the present paper, first a general algorithm for maximizing a differentiable goal function depending on a set of orthogonal matrices is formulated and then specified to the problem of estimating orthonormal transformation matrices for rotating core arrays to simpler structure. The generality of the chosen approach allows to cope with all possible transformation criteria by just changing one command in the implementation. In particular, the classical body-and slice-wise diagonalization of core arrays as well as the recently proposed maximization of the variance of squared entries are covered. The stability of the algorithm is addressed by a simulation study using 120 three-way core arrays of dimension (4,4,4). Each core array instantiates a class of 50 equivalent cores by random orthonormal transformations. Theoretically, each core within a given class has the same optimum with respect to the chosen criterion, and the ability of the algorithm to provide that result has been investigated. The algorithm proves to work with a high degree of stability and consistency in optimizing the three discussed goal functions. In addition, theoretical convergence results of the algorithm are provided. In particular, monotonic convergence of functional values and convergence of iterates towards a stationary solution an proven. To illustrate the effect of maximizing the variance-of-squares and the functionality of the algorithm, the proposed method is applied to a three-way data array from fluorometric analysis of fractions obtained from low-pressure chromatographic separation of a preliminary sugar product, thick juice. A significant gain in simplicity is achieved, and in particular optimizing variance-of-squares provides a simple core structure for the data under investigation. The proposed algorithms for maximizing variance-of-squares, body diagonality and slice-wise diagonality have been implemented in MATLAB and are available by contact to the authors. (C) 1999 Elsevier Science B.V. All rights reserved.